Factoring a Polynomial

It is second-order because the highest power of is (only
non-negative integer powers of are allowed in this context). The
polynomial is also
monic
because its leading coefficient, the
coefficient of , is . By the fundamental theorem of algebra
(discussed further in §2.4), there are exactly two
roots
(or
zeros) of any
second order polynomial. These roots may be real or complex (to be defined).
For now, let's assume they are both real and denote them by
and . Then we have
and , and we can write

This is the factored form of the monic polynomial .
(For a non-monic polynomial, we may simply divide all coefficients
by the first to make it monic, and this doesn't affect the zeros.)
Multiplying out the symbolic factored form gives

Comparing with the original polynomial, we find we must have

This is a system of two equations in two unknowns. Unfortunately, it is a
nonlinear system of two equations in two
unknowns.2.1 Nevertheless, because it is so small,
the equations are easily solved. In beginning algebra, we did them by
hand. However, nowadays we can use a software tool such as Matlab or
Octave to solve very large systems of linear equations.

The factored form of this simple example is

Note that polynomial factorization rewrites a monic th-order
polynomial as the product of first-order monic polynomials,
each of which contributes one zero (root) to the product. This
factoring business is often used when working with digital
filters [68].

The Quadratic Formula

The general second-order (real) polynomial is

(2.1)

where the coefficients
are any real numbers, and we assume since otherwise
it would not be second order. Some experiments plotting for different
values of the coefficients leads one to guess that the curve is always a
scaled and translated parabola. The canonical parabola centered
at is given by

(2.2)

where the magnitude of determines the width of the parabola, and
provides an arbitrary vertical offset. If , the parabola has
the minimum value at ; when , the parabola reaches a
maximum at (also equal to ). If we can find in
terms of for any quadratic polynomial, then we can easily
factor the polynomial. This is called completing the square.
Multiplying out the right-hand side of Eq.(2.2) above, we get

(2.3)

Equating coefficients of like powers of to the general second-order
polynomial in Eq.(2.1) gives

Using these answers, any second-order polynomial
can be rewritten as a scaled, translated parabola

In this form, the roots are easily found by solving to get

This is the general quadratic formula. It was obtained by simple
algebraic manipulation of the original polynomial. There is only one
``catch.'' What happens when is negative? This introduces the
square root of a negative number which we could insist ``does not exist.''
Alternatively, we could invent complex numbers to accommodate it.

Complex Roots

Figure 2.1:
An example parabola defined by
.

As a simple example, let , , and , i.e.,

As shown in Fig.2.1, this is a parabola centered at (where
) and reaching upward to positive infinity, never going below .
It has no real zeros. On the other hand, the quadratic formula says that the
``roots'' are given formally by
. The
square root of any negative number can be expressed as
, so the only new algebraic object is .
Let's give it a name:

Then, formally, the roots of are , and we can formally
express the polynomial in terms of its roots as

We can think of these as ``imaginary roots'' in the sense that square roots
of negative numbers don't really exist, or we can extend the concept of
``roots'' to allow for complex numbers, that is, numbers of the form

It can be checked that all algebraic operations for real
numbers2.2 apply equally well to complex numbers. Both real numbers
and complex numbers are examples of a
mathematical field.2.3 Fields are
closed with respect to multiplication and addition, and all the rules
of algebra we use in manipulating polynomials with real coefficients (and
roots) carry over unchanged to polynomials with complex coefficients and
roots. In fact, the rules of algebra become simpler for complex numbers
because, as discussed in the next section, we can alwaysfactor
polynomials completely over the field of complex numbers while we cannot do
this over the reals (as we saw in the example
).

Fundamental Theorem of Algebra

This is a very powerful algebraic tool.2.4 It says that given any polynomial

we can always rewrite it as

where the points are the polynomial roots, and they may be real or
complex.

This section introduces various notation and terms associated with complex
numbers. As discussed above, complex numbers arise by introducing
the square-root of as a primitive new algebraic object among real
numbers and manipulating it symbolically as if it were a real number
itself:

Mathematicians and physicists often use instead of as .
The use of is common in engineering where is more often used for
electrical current.

As mentioned above, for any negative number , we have

where
denotes the absolute value of . Thus, every square
root of a negative number can be expressed as times the square
root of a positive number.

By definition, we have

and so on. Thus, the sequence
,
is a
periodic sequence with period, since
. (We'll
learn later that the sequence is a sampled complex sinusoid having
frequency equal to one fourth the sampling rate.)

Every complex number can be written as

where and are real numbers.
We call the real part and
the imaginary part.
We may also use the notation

Note that the real numbers are the subset of the complex numbers having
a zero imaginary part ().

The rule for complex multiplication follows directly from the definition
of the imaginary unit :

In some mathematics texts, complex numbers are defined as ordered pairs
of real numbers , and algebraic operations such as multiplication
are defined more formally as operations on ordered pairs, e.g.,
. However, such
formality tends to obscure the underlying simplicity of complex numbers as
a straightforward extension of real numbers to include
.

It is important to realize that complex numbers can be treated
algebraically just like real numbers. That is, they can be added,
subtracted, multiplied, divided, etc., using exactly the same rules of
algebra (since both real and complex numbers are mathematical
fields). It is often preferable to think of complex numbers as
being the true and proper setting for algebraic operations, with real
numbers being the limited subset for which .

The Complex Plane

Figure 2.2:
Plotting a complex number as a point in the complex plane.

We can plot any complex number in a plane as an ordered pair
, as shown in Fig.2.2. A complex plane (or
Argand diagram) is any 2D graph in which the horizontal axis is
the real part and the vertical axis is the imaginary
part of a complex number or function. As an example, the number
has coordinates in the complex plane while the number has
coordinates .

Plotting as the point in the complex plane can be
viewed as a plot in Cartesian or
rectilinear coordinates. We can
also express complex numbers in terms of polar coordinates as
an ordered pair
, where is the distance from the
origin to the number being plotted, and is the angle
of the number relative to the positive real coordinate axis (the line
defined by and ). (See Fig.2.2.)

Using elementary geometry, it is quick to show that conversion from
rectangular to polar coordinates is accomplished by the formulas

where
denotes the arctangent of (the angle
in radians whose tangent is
), taking the
quadrant of the vector into account. We will take in
the range to (although we could choose any interval of
length radians, such as 0 to , etc.).

In Matlab and Octave, atan2(y,x) performs the
``quadrant-sensitive'' arctangent function. On the other hand,
atan(y/x), like the more traditional mathematical notation
does not ``know'' the quadrant of , so it maps
the entire real line to the interval
. As a specific
example, the angle of the vector
(in quadrant I) has the
same tangent as the angle of
(in quadrant III).
Similarly,
(quadrant II) yields the same tangent as
(quadrant IV).

The formula
for converting rectangular
coordinates to radius , follows immediately from the
Pythagorean theorem, while the
follows from the definition of the tangent
function itself.

Similarly, conversion from polar to rectangular coordinates is simply

These follow immediately from the definitions of cosine and sine,
respectively.

More Notation and Terminology

It's already been mentioned that the rectilinear coordinates of a complex
number in the complex plane are called the real part and
imaginary part, respectively.

We also have special notation and various names for the polar
coordinates of a complex number :

The complex conjugate of is denoted
(or ) and is defined by

where, of course,
.

In general, you can always obtain the complex conjugate of any expression
by simply replacing with . In the complex plane, this is a vertical flip about the real axis; i.e., complex conjugation
replaces each point in the complex plane by its mirror image on the
other side of the axis.

Since
is the algebraic expression of in terms of its
rectangular coordinates, the corresponding expression in terms of its polar
coordinates is

There is another, more powerful representation of in terms of its
polar coordinates. In order to define it, we must introduce Euler's
identity:

(2.5)

A proof of Euler's identity is given in the next chapter.
Before, the only algebraic representation of a complex number we had was
, which fundamentally uses Cartesian (rectilinear) coordinates in
the complex plane. Euler's identity gives us an alternative
representation in terms of polar coordinates in the complex plane:

We'll call
the polar form of the complex number
, in contrast with the rectangular form. Polar
form often simplifies algebraic manipulations of complex numbers,
especially when they are multiplied together. Simple rules of
exponents can often be used in place of messier trigonometric
identities. In the case of two complex numbers being multiplied, we
have

A corollary of Euler's identity is obtained by setting
to get

This has been called the ``most beautiful formula in mathematics'' due
to the extremely simple form in which the fundamental constants
, and 0, together with the elementary operations of addition,
multiplication, exponentiation, and equality, all appear exactly once.

For another example of manipulating the polar form of a complex number,
let's again verify
, as we did above in
Eq.(2.4), but this time using polar form:

As mentioned in §2.7, any complex expression can be conjugated
by replacing by wherever it occurs. This implies
,
as used above. The same result can be obtained by using Euler's
identity to expand
into
and negating the imaginary part
to obtain
,
where we used also the fact that cosine is an even function
(
) while sine is odd
(
).

We can now easily add a fourth line to that set of examples:

Thus,
for every .

Euler's identity can be used to derive formulas for sine and cosine in
terms of
:

Similarly,
, and
we obtain the following classic identities:

De Moivre's Theorem

As a more complicated example of the value of the polar form, we'll prove
De Moivre's theorem:

Manipulations of complex numbers in Matlab and Octave are illustrated
in §I.1.

To explore further the mathematics of complex variables, see any
textbook such as Churchill [15] or LePage [37].
Topics not covered here, but which are important elsewhere in signal
processing, include analytic functions, contour integration, analytic
continuation, residue calculus, and conformal mapping.